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How many numbers between 1 and 200

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Re: How many numbers between 1 and 200 [#permalink]
chetan2u wrote:
GoonerForever wrote:
How many numbers between 1 and 200 are exactly divisible by exactly two of 3, 9 and 27?

a. 14
b. 15
c. 16
d. 17
e. 18

Anything divisible by 27 will be div by 3 and 9 also...
Anything divisible by 9 is also divisible by 3,
SO we look for multiple of 9 till 200, but these should not be divisible by 27..

Multiples of 9 till 200 = 200/9=22.xy, so 22 multiples
Multiples of 27 till 200= 200/27=7.xy, so 7 multiples

Can you elaborate more on this from scratch?
Thank you

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Re: How many numbers between 1 and 200 [#permalink]
We have the multiples of three number which are 3, 9 and 27.
The logic behind this problem is obvious:
One should understand the all multiples of 27 also satisfy 9 and all multiples of 9 also satisfy 3.
In order to find the multiples of exactly two of these number, we should exclude the third number. Here we should think about what number to exclude to have the exact multiples of two of them.
There can be three cases:
So, let's firstly choose the multiples of 27 and 3. One can notice that all multiples of 27 satisfy the multiples of 9, so it's illogical to choose the exact multiples of 27 and 3 excluding the multiples of 9, because it also excludes the multiples of 27.
For the second case, to choose 9 and 27 is also an illogical decision. All multiples of 9 and 27 satisfy the multiples of 3.
Here remains the third case: 9 and 3. The multiples of 9 satisfy the multiples of 3, but they also the multiples of 27 are in the list of 9's multiples, such as 27, 54,81.
Therefore, to get the exact multiples of two number, we should find the total number of multiples of 9, then subtract the multiples of 27 from that to get the exact multiples of 9 and 3.
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Re: How many numbers between 1 and 200 [#permalink]
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Re: How many numbers between 1 and 200 [#permalink]
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